


On the Convergence of Interpolating Polynomials Using Chebyshev

by Bitenomnom



Series: Mathematical Proof [44]
Category: Sherlock (TV)
Genre: Gen, John does casework, John is kidnapped, Johnlock goggles optional, M/M, Mathematics, Post-Reichenbach, Sherlock and John locked in a restroom stall together, Sherlock infiltrates crime rings, lips on ears, lips on hands
Language: English
Status: Completed
Published: 2012-11-28
Updated: 2012-11-28
Packaged: 2017-11-19 18:06:12
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,010
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/576145
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>John finds out that Sherlock is alive.<br/>John finds out that Sherlock is alive much, much earlier than Sherlock was planning.</p>
            </blockquote>





	On the Convergence of Interpolating Polynomials Using Chebyshev

**Author's Note:**

> Soooo, I was going to say that I didn't mean for this to come off like the first chapter of something bigger, but then I remembered that this is something I keep thinking I'd like to write a multi-chapter fic about that I've had stewing in my head for a couple-few months now. XD I make no apologies! But I may, in the future, write an actual entire story about this. (Not promising anything, though.) Maybe at least I can do a follow-up within this series. For now, I hope this is enough.
> 
> As it says in the tags: Johnlock goggles optional. (But good luck taking them off once Sherlock starts getting all up in John's ear; I know I couldn't.)
> 
> By the way, I just realized that I only have six days of class left, and then it's finals week. Eep! So I am especially busy, but I also really want to make these last ones count -- bit conflicted. And, goodness. Can I just say right now how amazingly fun this has been? I wasn't expecting more than roughly two people to ever read these, so, it's been fantastic getting feedback from all of you and seeing people enjoy these quick little blurbs. I have a lot more to say, actually, but I'll save it for next week...essentially: sa;ljksadf;ljsafd, all of you are brilliant, thank you so much. You guys, by supporting me with your comments and even just with your presence, have turned this from a chore to the highlight of my day. I love you all!

When you create an interpolating polynomial (see [Interpolation](569158)), you use a certain number of nodes, or points. These points can be selected in various ways. For instance, they can be part of a uniform mesh (they are equally spaced along the interval you’re concerned with), or they can be chosen by finding the roots of the Chebyshev polynomial of the appropriate degree. The Chebyshev polynomial comes from a special (and really cool) recursive formula (look it up if you’re interested!).

Now, one might wonder if increasing the number of points will necessarily mean that the error between the function and the polynomial approximating it will go to zero. If you use evenly spaced points, it doesn’t necessarily do so! However, if you choose your points using the roots of the Chebyshev polynomial, the error does converge to zero as the number of points gets bigger and bigger. In other words, it is possible to choose some set of points in a range in a way that increasing the number of points decreases the upper bound of the error (the error converges to zero, or, your polynomial converges to the function!).

***

            Perhaps it was a simple matter of inevitability.

            There were only so many places in London, after all, and Sherlock was all the fool more for hanging about.

            The way it happened was unexpected, at least: for his part, Sherlock had made plenty of effort not to be seen, to make sure he visited places John would never visit, except for those few times he’d peered in on John’s life intentionally, cautiously, ready to disappear at a moment’s notice. John frequented the usual places: Baker Street—though now he mainly only came by to visit Mrs. Hudson, living in his own flat after Sarah hired him back—the grocery, where he occasionally ventured out of his relative hermithood in his flat—and, once or twice, a nearby bar where he met up with some old friend or another, Mike, Murray.

            John also visited Sherlock’s grave.

            But that was about it, as far as Sherlock could tell. Work, home, work, home, groceries, home, out with Mike Stamford, home, work, home, Sherlock’s grave, home, home, home, home.

            Well.

            “Home.”

            Of course it wasn’t John’s home.

            Sherlock could tell by his stiff shoulders and stiffer gait that John wasn’t happy there—wasn’t happy at all, really. But he was getting by, and that was all Sherlock needed, for now, for the next few months, or the next year, or however long it took to dismantle Moriarty’s web, ever-regenerating, self-sufficient in so many ways that cutting off just one connection was never enough. As long as John could get by, that was fine, and then Sherlock would come back, and hope John got better—at the very least, he could decide he hated Sherlock beyond words and move on with his life. None of this constant and quiet sobriety.

            Sherlock wasn’t just in London to peer in on John: He had business here, business that he felt much more comfortable taking up personally than trusting Mycroft or any of his men to handle. In fact, in this latest venture to London, some four months after he’d faked his death, Sherlock had managed to keep himself from checking in on John entirely.

            He’d found some dingy motel to stay the night out, and made his way to an equally dingy club next door—just the sort of place his next target might have friends.

            But he’d had no opportunity to find such friends there, and that…

            That was because he found John first.

 

 

 

            Greg hadn’t really put John up to it, but they’d gone out and tried to enjoy a football game together and Greg had mentioned it, so John had volunteered.

            “Thanks, mate,” Greg had said, and, “Should hire you onto the force,” at which John had laughed sharply and bitterly before managing to tone it down to a chuckle.

            According to Greg, they’d been looking for this guy for a while.

            “What for? I mean, what’d he do?”

            “That’s the thing,” he’d said, “we don’t really know. All we know is, he’s somehow involved in up to about twenty different crime rings, and he’s very good at hiding.”

            “So you don’t have a name, or…anything?”

            “Nothing.”

            “So how’m I meant to find him?” John leaned forward, feeling more awake and more alive than he had in the past few months. Maybe he wouldn’t be able to figure it out at all, but it was a fair sight more interesting than his GP work, and he really needed to get out a bit—even if it was to what Greg assured him was the stickiest, seediest club he could envision.

            He’d bring his gun. It could be fun.

            Greg had chewed thoughtfully on his chips. “Well,” he started, “we think he’s at this place to make contacts. Got a boss or a co-conspirator or something that goes there a lot. So maybe just…listen for coded language, you know, keep an eye out for covert gestures. Dunno if he’ll look it, but if he’s gone this long and not been caught, he’ll be clever.”

            John thought of the cabbie. He heard a voice in his head question whether being able to avoid Scotland Yard really made one _clever_ or merely _not a complete idiot_ , but he bit back his initial impulse to relay the comment to Greg.

            “He’s likely to recognize anyone on our force,” Greg continued. “So you’re doing me a big favor, here.”

            “Mm,” John agreed. He wondered how true that was—after all, he’d been featured in a couple of newspaper photos himself—but that was months and months ago, and no one would assume he was going undercover on a case by himself, anyway.

            And so the next evening, John made his way to the club, keeping his eyes and ears out for anything that might be of importance. Conversations at the bar were woefully devoid of excitement, but he couldn’t hear anything any closer to the music, so he took a seat at the bar and ordered a pint. The gay couple beside him was bickering about how many drinks was too many; the woman who sat two seats away from him was interested, but too shy to say anything, and diverted her attention to speaking on her mobile to a friend about going to the midnight opening of a film, clearly surprised and displeased by the rather unsanitary state of the place.

            John rested his head in one hand and ran one finger around the rim of his emptied glass, chancing occasional glances to one side or the other, twisting around under the pretense of rubbing at his shoulder. He ordered another drink and got started on that one, and moved to a new seat on the other side of the array of chairs and tables and barstools not currently being occupied by university kids or druggies or quiet drunks. When he glanced to his left to make sure he wasn’t going to be invading someone’s space when he set his drink down, something caught in his throat.

            Someone was turned away from him, apparently glancing over the crowd, surveying the area.

            John tapped his shoulder, too afraid to say the name lest he was wrong, because he definitely _was_ wrong, because there was absolutely _no_ way that…

            The man turned around.

            “Sherlock?” John mouthed, because it was. It was. It _was._ No, he was probably just drunker than he thought. He’d just ordered two beers, hadn’t he? That couldn’t be it, then. Well, maybe he was high. Maybe the air in this place was somehow laced with almost as many drugs as the people in it were.

            Sherlock’s—if that’s who he was—eyes widened, apparently in panic. He looked ready to dash off, and so John grabbed him by the wrist, and it was definitely there, definitely solid, and this _had_ to be Sherlock, because nobody could possibly look and feel and act this much like Sherlock without being him.

            “You’re hallucinating,” he said.

            “I don’t think I am,” John answered, and held tighter.

            “What are you doing here?”

            “We’re having this conversation somewhere else,” John said, suddenly angry, frightfully angry, nails nearly piercing into Sherlock’s skin, because if this was Sherlock, and it _was_ , then Sherlock was alive, somehow, and Sherlock was _alive_ and John didn’t know, or hadn’t known before, at least, and fuck if John was just going to let _that_ one slip by him.

            Sherlock, thankfully, seemed to find the idea agreeable, shooting furtive glances around the place. He nodded toward the back corner—the restroom.

            “I was thinking outside,” John said.

            “We can’t.”

            John’s grip tightened, if such a thing was possible. Sherlock hopped off his chair and tried to tug John with him, and, grudgingly, John followed.

            “In,” Sherlock motioned to a stall. John gave him a wary glance and entered, and Sherlock, without a word, stepped onto the seat of the toilet, feet lined up as well as they could be with its open ring, and squatted so that he did not peer over the walls of the stall. “Face forward,” he mouthed, and then pulled John back toward him and whispered in his ear at what couldn’t have qualified as more than half a decibel, “we may be here a while. Nod yes if you understand.”

            John nodded, and realized with crushing disappointment that if this was Sherlock’s arrangement, he wouldn’t be able to ask the man any questions at all—trivial matters like, “How the hell are you alive?” or “What are you doing?” or “Do you have any idea what I ought to do to you for doing this to me?”

            “I had to do it, John,” Sherlock continued, just as quietly as before, his lips against John’s ear so that John could hear him. Sherlock placed a hand on either of John’s shoulders, either to help him keep his balance or to keep John in place, or both. “I’m tearing down his web, and I have to do it in secret. They _have_ to believe I am dead.”

            John nodded, swallowed, balled and unballed his fists, still trying to work past the fact that up until two minutes ago he also believed Sherlock dead, and maybe still did, a little.

            “They would never believe it if you knew, because your,” he paused, seemed hesitant to say it, to acknowledge it, “mourning would be markedly not genuine. These are people who make their entire lives on identifying liars.”

            “Right, and...” John started saying, and Sherlock clapped a hand over his mouth.

            “He was going to kill you,” Sherlock mouthed onto John’s ear. “If I didn’t jump.”

            John shivered. “What now?” he moved his lips against Sherlock’s hand.

            “You can’t go back,” Sherlock answered after a time. “They’ll know something’s different and take you hostage to try to extract the information themselves.” At a stiffening of John’s posture, Sherlock said, “Yes; they’re monitoring you. Maybe not at this very instant, but they check in, I’m positive.”

            He seemed deep in thought.

            “MacDermot,” Sherlock finally muttered. John’s brows creased. “Yes, we can use him. He’ll kidnap you.” John could feel him shift slightly. “It will catch Lestrade’s attention, but he likely won’t be surprised—he sent you here, didn’t he?—he may hunt with greater fervor, but he won’t find MacDermot—or you. Not for a while, anyway.”

            “What?” John mouthed.

            “You might know him as Neeman or Lou or Chebyshev or Gilbert or Reed, or maybe just as that bloke who seems to be involved in every crime ring this side of the Atlantic.”

            John nodded. The person Greg had told him about—was he one of Sherlock’s contacts? Or was Sherlock suggesting—“Who is he?” John finally whispered, his breath echoing against Sherlock’s hand with a muffled hiss that could be mistaken for water in the pipes.

            He could practically hear Sherlock smirking as his breath rushed around the shell of John’s ear. “He’s me.”

            


End file.
